Another field contruction

Construct a field of order 125.

I don't know, this chapter is just not coming to me for some reasons, I can't really understand how to construct a field. So I pick a field, then pick a poly that is irreducible and has order of 125, how do I do that?

I don't know, this chapter is just not coming to me for some reasons, I can't really understand how to construct a field. So I pick a field, then pick a poly that is irreducible and has order of 125, how do I do that?

Again remember what I said. Let be a finite field . Let be an irreducible polynomial over . Then is a field. Now any element in can be written uniquely as now for (for ) we have choices for the coefficients. Thus, in total there are such elements in this larger field.

1)Let
2)Let be an order irreducible polynomial in .
3)The factor ring is a field with elements.

So the thing remaining now is for you to find an irreducible degree polynomial.